\[ \begin{array}{l} p(x)=2 x+2 \\ q(x)=-x^{2}-2 \end{array} \] Find the following. \[ \begin{array}{l} (p \circ q)(4)= \\ (q \circ p)(4)= \end{array} \]
Solution
Step 1 ：Define the functions \(p(x)=2x+2\) and \(q(x)=-x^2-2\).
Step 2 ：To find \((p \circ q)(4)\), we first need to find \(q(4)\) and then substitute that result into \(p(x)\).
Step 3 ：Calculate \(q(4)\) by substituting \(x=4\) into \(q(x)\), which gives us \(-4^2-2=-18\).
Step 4 ：Substitute \(-18\) into \(p(x)\) to get \(p(-18)=2*(-18)+2=-34\). So, \((p \circ q)(4)=-34\).
Step 5 ：To find \((q \circ p)(4)\), we first need to find \(p(4)\) and then substitute that result into \(q(x)\).
Step 6 ：Calculate \(p(4)\) by substituting \(x=4\) into \(p(x)\), which gives us \(2*4+2=10\).
Step 7 ：Substitute \(10\) into \(q(x)\) to get \(q(10)=-10^2-2=-102\). So, \((q \circ p)(4)=-102\).
Step 8 ：Final Answer: \((p \circ q)(4) = \boxed{-34}\), \((q \circ p)(4) = \boxed{-102}\)
